Calculation Methods for Wallenius' Noncentral Hypergeometric Distribution

Transcription

1 Clulo Mehos for Wlleus' oerl Hypergeoer sruo Ager Fog A revse verso of hs rle wll pper Couos Sss Sulo Copuo vol. 37 o. 8. SUMMARY. Two ffere proly sruos re oh ow he lerure s "he" oerl hypergeoer sruo. Wlleus' oerl hypergeoer sruo e esre s ur oel wh s. Fsher's oerl hypergeoer sruo s he ool sruo of epee ol vres gve her su. Severl ffere ehos for lulg proles fro Wlleus' oerl hypergeoer sruo re erve. Rge of pplly uerl proles effey re susse for eh eho. Approos o he e vre re lso susse. Ths sruo hs por pplos oels of se splg oels of evoluory syses. KEY WORS: oerl hypergeoer sruo Wlleus Fsher Mulvre sruo Proly fuo.. Irouo Two ffere proly sruos re oh ow he lerure s he oerl hypergeoer sruo. These wo sruos wll e lle Wlleus' Fsher's oerl hypergeoer sruo respevely. The oelure proles re susse elow. Fsher's oerl hypergeoer sruo s he ool sruo of epee ol vres gve her su MCullgh eler 983. Wlleus' oerl hypergeoer sruo s sruo of se splg. I e esre s ur oel whou replee wh s. Wlleus' sruo hs y poel pplos lug oels of seleve survvl seleve preo eology evoluory ology Mly 985 oels of ve effy Heráez-Suárez Csllo-Chvez s well s geerl oels of se splg Wlleus 963. The pplo of hs sruo hs ee hpere y he f h he oly pulshe lulo eho Lyos 98 s uerlly usle effe pplle oly o rrow rge of preers s eple elow. The purpose of he prese suy s o see relle lulo ehos h re pplle o we rge of preers lug he ulvre se. Mehos for splg fro hs sruo re esre opyg pper Fog 7.. efo properes Assue h ur os lls of ffere olors where s he uer of lls of olor C {... }. lls re sple oe y oe fro he ur whou replee suh wy h he proly h prulr ll s sple gve rw s proporol o propery w whh we wll ll wegh or os. The wegh of ll epes oly o s olor. Le X ν X ν X ν... X ν eoe he ol uer of lls of eh olor sple he frs rws. The proly h he e rw gves ll of olor s

2 X X p ν ν ν ν X. The proly fuo for hs sruo hs ee erve y Wlleus 963 for he uvre se y Chesso 976 for he ulvre se: whypg Ι Λ where Λ 3 Ι / whh s vl for >. The uepee egrl 4 rses s he soluo o sree fferee equo Wlleus 963 Chesso 976. The os e rrrly sle: whypg whypg : r r >. 7 The uvre sruo e efe s he proly fuo } whypg{ whypg 8 where - - w w w. The followg properes of he uvre sruo re esly erve: / whypg whypg w. 9. whypg whypg whypg. whypg whypg whypg The sruo of he lls h re lef he ur s o Wlleus' oerl hypergeoer sruo. Ths s l of syery h sgushes Wlleus' fro Fsher's oerl

3 hypergeoer sruo. We wll herefore efe he opleery Wlleus' oerl hypergeoer sruo s he sruo of he lls h re he ur: whypg whypg... w >. Ths fuo s use for oelg he sruo of survvors fer rw seleo proess Mly oelure prole The wo oerl hypergeoer sruos re ofe ofuse he lerure or ssue o e el e.g. Lyos 98 SAS Isue. sussos wh uer of relev sess hs le e o he oluso h he es soluo o he e ofl s o pply he prefes Wlleus' Fsher's o he e orer o sgush he wo sruos. Fsher's oerl hypergeoer sruo roue whou e y Fsher 935 ws frs gve he e eee hypergeoer sruo Hress 965 u soe of y orrespoes were srogly oppose o usg hs e. Whle he use of prefes es he es rher log hs he vge of ephszg h here s ore h oe oerl hypergeoer sruo wherey he rs of ofuso s ze. 4. Me vre The e µ... of X e pproe y ν µ ν µ ν µ ν ν pν µ ν µ. 3 µ Ths se of fferee equos e pproe y se of fferel equos wh he soluo gve y Mly 974: / / / µ µ µ... µ C : µ. 4 The soluo µ µ... µ s pproo o he e µ of X whh s vl uer he oos h C : > >. Ieresgly he e gve y 4 s goo pproo os ses eer h he vlue oe y ero of 3. 4 s e whe ll w re equl whle 3 s o. Mly 974 erve 4 for he purpose of esg he w 's fro eperel sples. To solve 4 for he es efe or Breslow Je Chesso y Jes Gele Crlos Herez-Surez or L. Johso Suel Koz Brue Lev Jgg Lo Bry Mly Peer MCullgh Jor Oñ Te Wlleus. 3

4 θ µ θ z / θ 5 solve zq y ewo-rphso ero. I he uvre se wh w > y e ore effe o solve µ µ. 6 Mly Mller Coo 97 gve e epresso for he vre s of he uvre sruo s fuo of... Uforuely hs epresso s so sesve o ures he es h s useless uless ll he es re ow wh very hgh preso. A pproo o he vre e oe y pprog Wlleus' oerl hypergeoer sruo wh Fsher's oerl hypergeoer sruo wh he se e usg pproe forul gve y Lev 984 for he vre of he ler sruo: σ σ F µ µ µ µ. 7 Ths pproo s goo whe w s er s fr fro. A sple reloshp ewee s he u of he proly fuo f s oe fro he orl sruo pproo: σ σ f M π 8 where M s he oe. Ths pproo s eer h s F soe ses where s s hgh. However 8 hs ovous lo se σ / π Mehos for lulo of he proly fuo Oly oe lulo eho hs hhero ee eoe he lerure Lyos 98. Uforuely hs eho s effe els serous uerl proles s eple elow. Severl oher ehos wll e evelope here he pplly of eh eho wll e susse. 5. Reursve lulo The os ovous lulo eho for he uvre sruo s he reursve pplo of. Fgure llusres hs eho. The fel wh oores represes he proly h here re ely lls of olor og he frs lls rw fro he ur. The vlue of eh fel s lule fro he fel o he lef of he fel elow he ler. The h of rrows osues rrry reory. Ths eho hs eelle uerl sly for ll preer vlues. The uulo of roug errors s o severe. uerl

5 5 uerflow our u sfely e gore. Ths eho s effe whe re hgh se he uer of proles o lule s -. The eooy of hs eho e prove y gorg eglgle proles fr fro he e. The reursve eho e use ll suos where he eooy of opuer resoures llows. Ths eho eoes ople effe he ulvre se. fg.. Reursve lulo of Wlleus proles. 5. Bol epso ehos Le r e posve sle for susue r o 4: Ι r r r. 9 Coser he uvre se le w w w r pply 3: whypg. Applyg he ol heore o - w swppg he orer of egro suo gves: whypg. Ths egrl s ow s he Be fuo Β hee

6 6 whypg Β. 3 Usg G! hs e reue o q whypg!! q 4 where he oo es he fllg forl power efe y. 5 The followg reurso forul hols for > >: q q. 6 As orollry of 4 he se gves whypg. 7 Ths eho e epe o he ulvre se y ol epso of he powers 9 for ll u he lrges of gvg - ese sus. I he se where ll u oe of he vlues re zero we ge { }..... whypg. 8 The suo 4 use serous uerl proles eve for oere vlues of he preers. For eple lulo of whypg46855 usg 4 wh oule preso gves he vlue ø The orre vlue s.53. Ths error s ue o loss of preso evee y he f h he uerlly lrges of he q ers s.3ÿ 9 es s lrge s h he fl su. Ths eho s herefore o relle uless he uer of suo ers s que sll. The eho gve y Lyos 98 s oe y ol epso of oh powers 4 for he uvre se. Ths gves whyp. 9 The uerl proles 9 re geerlly worse h 4. A lulo of he se uerl eple s ove wh hs eho gves he soewh uous resul -6.4ÿ. The error s ue o loss of preso oh sus. For he ouer su he uerlly lrges er s 7.ÿ 6 es he fl su. Furherore 9 s less eool h 4 euse of he ese

7 sus. Lyos' eho s herefore o reoele. 5.3 Tylor epso ehos Coser he efe egrl of rrry fuo: I Φ. 3 efe ulry fuo U orreo fuo Φ / U. Assue h hese hree fuos re ll ly he ople s { Â < R}. Epg Y Tylor seres swppg he orer of egro suo gves I U. 3! Ths epso s overge for < R euse egro oes o hge he rus of overgee. If U s syer rou he he o ers he su vsh: I U! U!. 3 The ulry fuo U shoul e hose so h hs egrl e lule lylly. The overgee of hs epso s os lely o e goo f U s hose so h Y hs os of s wegh er. A Tylor eho s o sue for he egrl 4 euse he egr hs os of s wegh er where os ses s o fferele. The rsfore egrl 9 s preferre. Le Φ r r r. 33 Logrhg ffereg gves r ϕ log Φ log r r log log 34 r r r r ϕ. 35 Alyss of ϕ shows h F hve sgle u he ervl < < whe r > /. The preferre vlue of he oe s ½. I orer o o hs vlue we efe z ϕ r r r r 36 7

8 oserve h he equo zr hs uque soluo he ervl / < r < whh e fou y he ewo-rphso ero: z r r f hs s > r z r r z r r r 7 oherwse r 8 log. 37 Heeforh we wll ssue h r s he soluo o zr so h he oe ½. Wh hese vlues of r we pply 3 o he lulo of he egrl 9. Three ffere hoes for he ulry fuo U wll e eplore: U 38 U A A Φ ϕ 39 { 4 } U A ϕ The preers of U U 3 re oh hose so h he frs wo ervves re equl o he ervves of F. The frs hoe U U gves I Φ. 4! The ervves F ½ re oe y logrh ffereo. Applyg Lez's rule for ffereo of prou o Φ Φ ϕ gves Φ. 4 Φ ϕ The 'h ervve of for > e epresse y he forul ζ ϕ 43 r! r r ζ ζ r ζ r r for for oherwse > < or lervely for > : η ϕ 44 r ϕ r 8

9 η η r η r r for for oherwse > < re oh prove y uo. The eres sgulres of F re eep he rre se h ll powers re egers. Therefore he rus of overgee s ½ he epso 4 s overge for < ½. A ypl shpe of he egr urve F s show fg.. I orer o ese he relve error whe he egro ervl s rrowe o < ½ we pproe he egr urve F wh he Guss urve U gve y 39. The relve error ue o he rrowe egro ervl s ε { Φ Φ } < Ι 45 erf Thus we f sule vlue of fro he esre preso ε:. erf ε. 46 The ypl se fg. where he Guss urve s poor pproo o F wll le o overeso of he relve error ε o he oluso h he Tylor eho s usule. fg.. Typl shpe of egr F w fg.. Iegr wh sore shpe w5 These resuls regrg overgee preso he vlue of lso per o he seo hr hoe for U susse elow. 9

10 The seo hoe U U gves! A I. 47 Se < we efe v susue θ v :! A I θ θ θ v v v. 48 Ths egrl s ow s he repee egrl of he error fuo Arowz Segu 965. Iegrg y prs es og h we hve. erf G 49 The ree epresso s resue of he ow seres erf. 5 Usg π!! we lule G!!!!. 5 Iserg 48 gves! G A I v. 5 To o he ervves of Y we efe log ψ. The frs wo ervves of re oe u log U. Therefore ψ ψ ψ for > ϕ ψ 3 3 ϕ ϕ 53 where s uersoo h he suo s zero whe he upper l s less h he lower l. The vlue of o ser 5 s v whh e splfe whe < ½ o

11 v erf ε. 54 The f h epes oly o he esre preso ε es vgeous o sore prelule vlues of G le for sule vlues of ε. The hr hoe U U 3 gves I A { 4 }. 55! Susug θ 4 gves 4 A A I Β 4 4! θ θ θ 56 4! where B y eoes he oplee Be fuo Β y α β y β α. 57 To o he ervves of Y we oe h he ervves of u log U e lule he se wy s he ervves of. Therefore he ervves of y - u e lule y g er er o he -su 44. For > η r ψ ψ 58 r where we efe / r. ow 3 ψ ψ Y½ Y'½ Y''½ A eperel evluo of he hree Tylor ehos esre here fs h he seo hr ehos hve oserly fser overgee h he frs eho. The overgee y e ssfory eve whe ½. I y ses he hr eho hs he fses overgee. everheless he seo eho y e preferre for eoo resos se use prelule vlues of G whe < ½ whle he hr eho requres he ore eosug lulos of he oplee Be fuo. The hr eho s preferre whe ½ where he oplee Be fuo s reple y he oplee Be fuo. To surze whypg e lule wh pproe preso ε f < ½ y usg equos or I s reoee o sle he su ers 5 wh A L o vo uer overflow uerflow. The evluos of he equos ll volve epressos of he ype y. Appropre Tylor epsos re reoee o vo loss of preso hese epressos whe y s er zero or lrge egve. The overgee y e poor whe s low. As for preg how goo he overgee s we efe he orlze reprol :

12 E. 6 A hgh vlue of E es sore egr urve E 5 for fg for fg. I hs ee fou eperelly h he overgee of he seo hr eho s goo whe <.5 E <. As orollry o he epso foruls we y o resole pproos y rug he seres. Trug he epso 56 o s frs er wh ½ gves he pproo π whypg Λ A. 6 3 Trug he epso 47 o s frs er wh gves he pproo π whypg Λ A. 6 The epoe he frs er of 47 s u whh s ' orer Tylor pproo o log F. Ths pproo eho s ow s Lple's eho resulg 6. Aorg o Beer Orszg 978:7 he ury of Lple's eho e prove y g ore ers o he Tylor epso of. The lulo org o Beer Orszg's eho volves ese Tylor sus possle overgee proles. These svges re voe here y epg Y rher h. Ths eho s provee o Lple's eho wh geerl pplly o egrls of uol fuos. 5.4 Coue fro epso The overgee of he oveeoe hree epsos e elere oserly y overso of he Tylor epsos o he orrespog oue fro epsos ses where s er or equl o ½ he overgee s poor. o provee s oe ses where he overgee of he Tylor epso s lrey goo. The oue fro epso orrespog o Tylor epso s lule y he eho esre y Perro 93. Whle he Tylor epso s heorelly overge for < ½ he orrespog oue fro epso s heorelly verge u prlly pplle. The oue fro eho hs he svges h requres he lulo of lrge Hel eers h s fful o evlue he preso oe. Ths eho wll herefore o e esre furher el here. 5.5 uerl egro uerl egro s eee ses where oe of he oveeoe lulo ehos re pplle. The egr 4 s o sue for uerl egro euse hs os of s wegh er where os ses s o ly. We prefer o egre F gve y 33 usg he vlue of r oe fro We y prove he perfore y egrg FF- over hlf he ervl o e vge of he f h F s los syer. A Guss-Legere eho Evs 995 wh 4 - pos vrle sep legh s sule. The sep legh shoul e sll where he egr urve s seepes whh s he epos or fleo pos s see fgure.

13 6. Sofwre pleeo A C pleeo of he ehos esre here e owloe fro 7. Suggesos for fuure reserh Beer pproos o he e vre woul e useful. uerl egro s he oly eho h s sule for he ses where E re oh hgh. A ore effe lulo eho overg suh ses s eee. Equo 46 gves pproe upper l o he relve error of he Tylor epso ehos. Ths error ese ppers o e relle pre u e upper l o he error y e preferle. Fsher's oerl hypergeoer sruo s ore well-reserhe h Wlleus' sruo. More reserh o he ehvor of he ler s eee. A oprso of he wo sruos hs ever ee pulshe. REFERECES Arowz M. Segu I. A Hoo of Mhel Fuos Wh Foruls Grphs Mhel Tles. ew Yor: ower. Beer C. M. Orszg S. A 978. Ave Mhel Mehos for Sess Egeers. ew Yor: MGrw-Hll. Chesso J A o-erl ulvre hypergeoer sruo rsg fro se splg wh pplo o seleve preo. Jourl of Apple Proly Evs G. A Prl uerl Alyss. Chheser: Joh Wley & Sos. Fsher R. A The Log of Iuve Iferee. Jourl of he Royl Ssl Soey Fog A. 7. Splg Mehos for Wlleus' Fsher's oerl Hypergeoer sruos. Worg pper. Hress W. L Properes of he Eee Hypergeoer sruo. Als of Mhel Sss Heráez-Suárez C. M. Csllo-Chvez C.. Ur oels ve effy eso. Sss Mee Lev B Sple Iprovees o Corfel's Approo o he Me of oerl Hypergeoer Ro Vrle. Boer Lyos. I. 98. Close Epressos for oerl Hypergeoer Proles. Couos Sss B: Sulo Copuo Mly B. F. J A Moel for Cer Types of Seleo Eperes. Boers Mly B. F. J The Sss of url Seleo o Al Populos. Loo: Chp Hll. Mly B. F. J. Mller P. Coo L. M. 97. Alyss of Seleve Preo Epere. The Aer urls MCullgh P. eler J. A Geerlze Ler Moels. Loo: Chp & Hll. Perro O. 93. e Lehre vo e Keerühe. Lepzg: B. G. Teuer. SAS Isue. SAS Oleo 9h e. Cry C: SAS Isue I. Wlleus K. T Bse Splg: The o-erl Hypergeoer Proly sruo. Ph.. hess Sfor Uversy Also pulshe wh he se le s Tehl repor o. 7. epre of Sss Sfor Uversy. 3